# Empty Set is Open in Metric Space

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## Theorem

Let $M = \struct {A, d}$ be a metric space.

Then the empty set $\O$ is an open set of $M$.

## Proof

By definition, an open set $S \subseteq A$ is one where every point inside it is an element of an open ball contained entirely within that set.

That is, there are no points in $S$ which have an open ball some of whose elements are not in $S$.

As there are no elements in $\O$, the result follows vacuously.

$\blacksquare$

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Theorem $6.4$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.10$